Which of the following graphs represents an exponential function?

• A. A downward-opening parabola with the equation $y=-x^2$
• B. A straight line with the equation $y=2x+1$
• C. A horizontal line with the equation $y=3$
• D. A graph that passes through the point $(0,1)$ and increases rapidly as $x$ increases, with the equation $y=2^x$

Answer: (D)

Verified step by step guidance

1. Recall that an exponential function has the form \(y = a^{x}\), where the variable \(x\) is in the exponent, and \(a\) is a positive constant not equal to 1.
2. Examine each given equation to identify which one fits the exponential form: - \(y = -x^{2}\) is a quadratic function because the variable \(x\) is raised to a power, not in the exponent. - \(y = 2x + 1\) is a linear function because \(x\) is to the first power and not in the exponent. - \(y = 3\) is a constant function, which is a horizontal line.
3. Look at the description of the correct graph: it passes through the point \((0,1)\) and increases rapidly as \(x\) increases. This matches the behavior of an exponential function because \(a^{0} = 1\) for any positive \(a\).
4. Identify the equation \(y = 2^{x}\) as the exponential function since the variable \(x\) is in the exponent and the base 2 is a positive constant greater than 1, which causes the rapid increase.
5. Conclude that the graph representing the exponential function is the one with the equation \(y = 2^{x}\), passing through \((0,1)\) and increasing rapidly as \(x\) increases.